A Memoir on Integral Functions. 121 



" A Memoir on Integral Functions." By E. W. BARNES, M.A., 

 Fellow of Trinity College, Cambridge. Communicated by 

 Professor A. R. FpRSYTH, Sc.D., F.lt.S. Received July 25, 

 1901. 



(Abstract.) 



The memoir deals with the asymptotic expansion of functions with 

 a single essential singularity at infinity in the neighbourhood of that 

 singularity. The term " integral function " is used as a translation of 

 the French expression " fonction entiere." 



Part I opens with an introduction, in which it is pointed out that for 

 each of the few integral functions whose detailed properties have been 

 investigated, there always exists near infinity an asymptotic expansion 

 valid in those parts of the region near infinity which are not at a finite 

 distance from zeros of the function. It is then suggested that wide 

 classes of integral functions admit such expansions ; and it is pointed 

 out that, if such a theorem can be proved and the expansions obtained, 

 we may solve many questions relating to the " genre " of a function, 

 to the nature of its zeros, and to the character of its coefficients when 

 expanded as a Taylor's series. We may, in fact, to a large extent 

 classify such functions by their behaviour at infinity. 



After the introduction, a short account of the historical development 

 of the enquiry is given ; and then the memoir proper commences with 

 a formal arrangement of integral functions according to the laws of 

 distribution of the zeros as seen in expressions in Weierstrassian- 

 product form. 



When the nth zero a n is such that it depends solely upon n and 

 certain definite constants, and also such that the law of dependence is 

 the same for all zeros, we call it a simple integral function with a single 

 sequence of non-repeated zeros, or sometimes briefly a simple integral 

 function. 



The zeros are said to be algebraic, when they are given, when n is 

 large, by a formula of the form 



fl + -^- + -=- + . . .1 , 

 ?iPi HP* 



where p is a rational positive quantity, and pi, p, . . . are rational 

 positive quantities arranged in ascending order of magnitude. If p 



CO J 00 | 



is such that 2 . - converges, and 1' , : diverges, however 



i I fl P + e i fl P e 



n=l | "ft | "=1 1 ' n I 



small the positive quantity may be, the function is said to be of 

 order p. 



We can form a scale of simple integral functions arranged according 



